In the last part of the proof, we will investigate the estimate for \(I^{\operatorname{ext}}(t)\) , which exerts some differences from those in [1]}, [2]} since the exponential stability does not hold anymore.
According to the action of the infinitesimal representation of Schrödinger-type representations \(\pi _{\lambda }\)  on the generators of the first layer of the Lie algebra \(\mathfrak {b}_n\)  such that
\(\mathrm {d}\pi _{\lambda }(X_j)=\sqrt{|\lambda |}\,\partial _{w_j}\ \ \mbox{and}\ \ \mathrm {d}\pi _{\lambda }(Y_j)=i\,\mathrm {sign}(\lambda )\sqrt{|\lambda |} \,w_j,\) 
